Use the interactive to graph the line that goes through the points [tex]$(-4,-2)$[/tex] and [tex]$(2,0)$[/tex].

The point [tex]$(a, 1)$[/tex] lies on the line you graphed. What is the value of [tex]$a$[/tex]?

[tex]$\square$[/tex]



Answer :

Let's solve this problem step-by-step:

1. Determine the slope of the line:

We have two points [tex]$(-4, -2)$[/tex] and [tex]$(2, 0)$[/tex]. The formula for the slope, [tex]\( m \)[/tex], between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Plugging in the coordinates, we get:

[tex]\[ m = \frac{0 - (-2)}{2 - (-4)} = \frac{0 + 2}{2 + 4} = \frac{2}{6} = \frac{1}{3} \][/tex]

So, the slope of the line is [tex]\( \frac{1}{3} \)[/tex].

2. Find the y-intercept of the line:

We can use the point-slope form of the line equation, [tex]\( y = mx + b \)[/tex], to determine the y-intercept [tex]\( b \)[/tex]. Substituting [tex]\( m = \frac{1}{3} \)[/tex] and one of the given points, say [tex]\((-4, -2)\)[/tex]:

[tex]\[ -2 = \frac{1}{3}(-4) + b \][/tex]

Solving for [tex]\( b \)[/tex]:

[tex]\[ -2 = -\frac{4}{3} + b \][/tex]

Add [tex]\(\frac{4}{3}\)[/tex] to both sides:

[tex]\[ -2 + \frac{4}{3} = b \][/tex]

Convert [tex]\(-2\)[/tex] to a fraction with a common denominator of 3:

[tex]\[ -\frac{6}{3} + \frac{4}{3} = b \][/tex]

This gives:

[tex]\[ b = -\frac{6}{3} + \frac{4}{3} = -\frac{2}{3} \][/tex]

Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].

3. Determine the value of [tex]\( a \)[/tex] such that the point [tex]\( (a, 1) \)[/tex] lies on the line:

We know the point [tex]\( (a, 1) \)[/tex] lies on the line, so we can use the line equation [tex]\( y = \frac{1}{3}x - \frac{2}{3} \)[/tex] and set [tex]\( y \)[/tex] to 1:

[tex]\[ 1 = \frac{1}{3}a - \frac{2}{3} \][/tex]

Add [tex]\(\frac{2}{3}\)[/tex] to both sides to isolate the term involving [tex]\( a \)[/tex]:

[tex]\[ 1 + \frac{2}{3} = \frac{1}{3}a \][/tex]

Convert 1 to a fraction with a common denominator of 3:

[tex]\[ \frac{3}{3} + \frac{2}{3} = \frac{1}{3}a \][/tex]

Simplify:

[tex]\[ \frac{5}{3} = \frac{1}{3}a \][/tex]

Multiply both sides by 3 to solve for [tex]\( a \)[/tex]:

[tex]\[ a = 5 \][/tex]

Therefore, the value of [tex]\( a \)[/tex] is [tex]\(\boxed{5}\)[/tex].